Entropy of automorphisms of II1-factors arising from the dynamical systems theory

Let a countable amenable group G act freely and ergodically on a Lebesgue space (X, mu), preserving the measure mu. If T is an element of Aut(X, mu) is an automorphism alpha (Gamma) of the equivalence elation defined by G then T can he extended to an automorphism alpha (Gamma) of the II1-factor M=L-infinity(X, mu) x G. we prove that if T commutes with the action of G then H(alpha (Gamma)) = h(T), where H(alpha (Gamma)) is the Connes-Stormer entropy of alpha (Gamma) and h(T) is the Kolmogorov Sinai entropy of T. We also, prove that For given s and t, 0 less than or equal to s less than or equal to t less than or equal to infinity. there exists a T such that h(T) = s and H(alpha (Gamma))= t. (C) 2001 Academic press.